leg. All pairs of isosceles Heronian triangles are given by rational multiples of and for coprime integers u and v with u > v and u + v odd. Heronian triangles whose perimeter is four times a prime It has been shown that a Heronian triangle whose perimeter is four times a prime is uniquely associated with the prime and that the prime is congruent to or modulo . Yui, P. and Taylor, J. S., "CRUX, Problem 2331, Solution" Memorial University of Newfoundland (1999): 185-186 It is well known that such a prime can be uniquely partitioned into integers and such that (see Euler's idoneal numbers). Furthermore, it has been shown that such Heronian triangles are primitive since the smallest side of the triangle has to be equal to the prime that is one quarter of its perimeter. Consequently, all primitive Heronian triangles whose perimeter is four times a prime can be generated by for integers and such that is a prime. Furthermore, the factorization of the area is where is prime. However the area of a Heronian triangle is always divisible by . This gives the result that apart from when and which gives all other parings of and must have odd with only one of them divisible by . Heronian triangles with rational angle bisectors If in a Heronian triangle the angle bisector of the angle , the angle bisector of the angle and the angle bisector of the angle have a rational relationship with the three sides then not only but also , and must be Heronian angles. Namely, if both angles and are Heronian then , the complement of , must also be a Heronian angle, so that all three angle-bisectors are rational. This is also evident if one multiplies: together. Namely, through this one obtains: where denotes the semi-perimeter, and the area of the triangle. All Heronian triangles with rational angle bisectors are generated by where are such that where are arbitrary integers such that and coprime, and coprime. Heronian triangles with integer inradius and exradii There are infinitely many decomposable, and infinitely many indecomposable, primitive Heronian (non-Pythagorean) triangles with integer radii for the incircle and each excircle. A family of decomposible ones is given by and a family of indecomposable ones is given by Heronian triangles as faces of a tetrahedron There exist tetrahedra having integer-valued volume and Heron triangles as faces. One example